Ideal Battery

• A battery has to do work to separate charge

• Work done by battery is given by

  $$ W_\text{battery}=\int^+_- \vec F\cdot \text d \vec l $$

• Work is done to overcome the $E$-field inside

  $$ \frac{W_\text{battery}}{q}=\int^+-\vec f\cdot \text d\vec l=-\int^+- \vec E\cdot \text d\vec l=\phi_+-\phi_-=\Delta\phi $$

Where $\Delta\phi$ is the potential difference in Volts

• Electromotive force (EMF) $\mathcal{E}=\oint\vec f\cdot \text d\vec l$ is the total potential change in Volts around a closed loop
• EMF is not a force

Motional potential difference

Consider pulling a metal bar through a $B$-field

• Charges experience a force $F=qvB$

• Charges repel unit balanced by $E$-field $qE=qvB$

• $E$-field between $a$ and $b$ gives

  $$ \Delta\phi_{ab}=El=vBl $$

Motional EMF

Consider forces on electrons around the loop

• $a\to b$ downward force, move down
• $b\to c$ downward force, no movement
• $c\to d$ no magnetic force
• $d\to a$ downward force, no movement

Electrons move opposite of current $I$

A change of $B$-field led to a current

• Consider the flux of magnetic field

$$ \Phi_m=\int_S\vec B\cdot\text d \vec A=Blx $$

Unit of magnetic flux: $1$ Weber = $1$ Tm$^2$

• Relate the change in flux to the loop velocity $v=\dot x$

$$ \frac{\text d\Phi_m}{\text dt}=-Blv $$

• Define induced EMF $\mathcal E$

$$ \mathcal E=-\frac{\text d \Phi_m}{\text d t} $$

Motional EMF examples

• A bar magnet approaching a stationary conducting loop
• A conducting loop shrinking (or growing) in a uniform magnetic field